Is there anywhere in NYC - preferably Manhattan, but even a borough -
to play racquetball without a gym membership? Just paying on a
per-session basis?

You can pay to play at the 92nd Y on a per day pay schedule. 

92nd Street Y
1395 Lexington Avenue
New York, NY 10128

Reservations & Rentals
Racquetball Court Reservations $40 / $20 members per hour
Advance reservations are available to guarantee court time and space
for a fee. (Members can use open racquetball courts without
reservations for free though precedence is given to those who reserve
time.) Non-members must also purchase a guest pass. For information
call 212.415.5700. (press 6 then #1 to speak with someone)

According to the man I spoke with it is $30 for a day pass and you can
play raquetball (not the $40 listed above)